Dmitrii Eliuseev

2 years ago

Creating Images on Your Local PC Using Stable Diffusion AI

Deep learning-based generative art is being researched. As usual, self-learning is better. Some models, like OpenAI's DALL-E 2, require registration and can only be used online, but others can be used locally, which is usually more enjoyable for curious users. I'll demonstrate the Stable Diffusion model's operation on a standard PC.

Let’s get started.

What It Does

Stable Diffusion uses numerous components:

A generative model trained to produce images is called a diffusion model. The model is incrementally improving the starting data, which is only random noise. The model has an image, and while it is being trained, the reversed process is being used to add noise to the image. Being able to reverse this procedure and create images from noise is where the true magic is (more details and samples can be found in the paper).
An internal compressed representation of a latent diffusion model, which may be altered to produce the desired images, is used (more details can be found in the paper). The capacity to fine-tune the generation process is essential because producing pictures at random is not very attractive (as we can see, for instance, in Generative Adversarial Networks).
A neural network model called CLIP (Contrastive Language-Image Pre-training) is used to translate natural language prompts into vector representations. This model, which was trained on 400,000,000 image-text pairs, enables the transformation of a text prompt into a latent space for the diffusion model in the scenario of stable diffusion (more details in that paper).

This figure shows all data flow:

Model architecture, Source © https://arxiv.org/pdf/2112.10752.pdf

The weights file size for Stable Diffusion model v1 is 4 GB and v2 is 5 GB, making the model quite huge. The v1 model was trained on 256x256 and 512x512 LAION-5B pictures on a 4,000 GPU cluster using over 150.000 NVIDIA A100 GPU hours. The open-source pre-trained model is helpful for us. And we will.

Install

Before utilizing the Python sources for Stable Diffusion v1 on GitHub, we must install Miniconda (assuming Git and Python are already installed):

wget https://repo.anaconda.com/miniconda/Miniconda3-py39_4.12.0-Linux-x86_64.sh
chmod +x Miniconda3-py39_4.12.0-Linux-x86_64.sh
./Miniconda3-py39_4.12.0-Linux-x86_64.sh
conda update -n base -c defaults conda

Install the source and prepare the environment:

git clone https://github.com/CompVis/stable-diffusion
cd stable-diffusion
conda env create -f environment.yaml
conda activate ldm
pip3 install transformers --upgrade

Download the pre-trained model weights next. HiggingFace has the newest checkpoint sd-v14.ckpt (a download is free but registration is required). Put the file in the project folder and have fun:

python3 scripts/txt2img.py --prompt "hello world" --plms --ckpt sd-v1-4.ckpt --skip_grid --n_samples 1

Almost. The installation is complete for happy users of current GPUs with 12 GB or more VRAM. RuntimeError: CUDA out of memory will occur otherwise. Two solutions exist.

Running the optimized version

Try optimizing first. After cloning the repository and enabling the environment (as previously), we can run the command:

python3 optimizedSD/optimized_txt2img.py --prompt "hello world" --ckpt sd-v1-4.ckpt --skip_grid --n_samples 1

Stable Diffusion worked on my visual card with 8 GB RAM (alas, I did not behave well enough to get NVIDIA A100 for Christmas, so 8 GB GPU is the maximum I have;).

Running Stable Diffusion without GPU

If the GPU does not have enough RAM or is not CUDA-compatible, running the code on a CPU will be 20x slower but better than nothing. This unauthorized CPU-only branch from GitHub is easiest to obtain. We may easily edit the source code to use the latest version. It's strange that a pull request for that was made six months ago and still hasn't been approved, as the changes are simple. Readers can finish in 5 minutes:

Replace if attr.device!= torch.device(cuda) with if attr.device!= torch.device(cuda) and torch.cuda.is available at line 20 of ldm/models/diffusion/ddim.py ().
Replace if attr.device!= torch.device(cuda) with if attr.device!= torch.device(cuda) and torch.cuda.is available in line 20 of ldm/models/diffusion/plms.py ().
Replace device=cuda in lines 38, 55, 83, and 142 of ldm/modules/encoders/modules.py with device=cuda if torch.cuda.is available(), otherwise cpu.
Replace model.cuda() in scripts/txt2img.py line 28 and scripts/img2img.py line 43 with if torch.cuda.is available(): model.cuda ().

Run the script again.

Testing

Test the model. Text-to-image is the first choice. Test the command line example again:

python3 scripts/txt2img.py --prompt "hello world" --plms --ckpt sd-v1-4.ckpt --skip_grid --n_samples 1

The slow generation takes 10 seconds on a GPU and 10 minutes on a CPU. Final image:

The SD V1.4 first example, Image by the author

Hello world is dull and abstract. Try a brush-wielding hamster. Why? Because we can, and it's not as insane as Napoleon's cat. Another image:

The SD V1.4 second example, Image by the author

Generating an image from a text prompt and another image is interesting. I made this picture in two minutes using the image editor (sorry, drawing wasn't my strong suit):

I can create an image from this drawing:

python3 scripts/img2img.py --prompt "A bird is sitting on a tree branch" --ckpt sd-v1-4.ckpt --init-img bird.png --strength 0.8

It was far better than my initial drawing:

The SD V1.4 third example, Image by the author

I hope readers understand and experiment.

Stable Diffusion UI

Developers love the command line, but regular users may struggle. Stable Diffusion UI projects simplify image generation and installation. Simple usage:

Unpack the ZIP after downloading it from https://github.com/cmdr2/stable-diffusion-ui/releases. Linux and Windows are compatible with Stable Diffusion UI (sorry for Mac users, but those machines are not well-suitable for heavy machine learning tasks anyway;).
Start the script.

Done. The web browser UI makes configuring various Stable Diffusion features (upscaling, filtering, etc.) easy:

V2.1 of Stable Diffusion

I noticed the notification about releasing version 2.1 while writing this essay, and it was intriguing to test it. First, compare version 2 to version 1:

alternative text encoding. The Contrastive LanguageImage Pre-training (CLIP) deep learning model, which was trained on a significant number of text-image pairs, is used in Stable Diffusion 1. The open-source CLIP implementation used in Stable Diffusion 2 is called OpenCLIP. It is difficult to determine whether there have been any technical advancements or if legal concerns were the main focus. However, because the training datasets for the two text encoders were different, the output results from V1 and V2 will differ for the identical text prompts.
a new depth model that may be used to the output of image-to-image generation.
a revolutionary upscaling technique that can quadruple the resolution of an image.
Generally higher resolution Stable Diffusion 2 has the ability to produce both 512x512 and 768x768 pictures.

The Hugging Face website offers a free online demo of Stable Diffusion 2.1 for code testing. The process is the same as for version 1.4. Download a fresh version and activate the environment:

conda deactivate  
conda env remove -n ldm  # Use this if version 1 was previously installed
git clone https://github.com/Stability-AI/stablediffusion
cd stablediffusion
conda env create -f environment.yaml
conda activate ldm

Hugging Face offers a new weights ckpt file.

The Out of memory error prevented me from running this version on my 8 GB GPU. Version 2.1 fails on CPUs with the slow conv2d cpu not implemented for Half error (according to this GitHub issue, the CPU support for this algorithm and data type will not be added). The model can be modified from half to full precision (float16 instead of float32), however it doesn't make sense since v1 runs up to 10 minutes on the CPU and v2.1 should be much slower. The online demo results are visible. The same hamster painting with a brush prompt yielded this result:

It looks different from v1, but it functions and has a higher resolution.

The superresolution.py script can run the 4x Stable Diffusion upscaler locally (the x4-upscaler-ema.ckpt weights file should be in the same folder):

python3 scripts/gradio/superresolution.py configs/stable-diffusion/x4-upscaling.yaml x4-upscaler-ema.ckpt

This code allows the web browser UI to select the image to upscale:

The copy-paste strategy may explain why the upscaler needs a text prompt (and the Hugging Face code snippet does not have any text input as well). I got a GPU out of memory error again, although CUDA can be disabled like v1. However, processing an image for more than two hours is unlikely:

Stable Diffusion 4X upscaler running on CPU © Image by author

Stable Diffusion Limitations

When we use the model, it's fun to see what it can and can't do. Generative models produce abstract visuals but not photorealistic ones. This fundamentally limits The generative neural network was trained on text and image pairs, but humans have a lot of background knowledge about the world. The neural network model knows nothing. If someone asks me to draw a Chinese text, I can draw something that looks like Chinese but is actually gibberish because I never learnt it. Generative AI does too! Humans can learn new languages, but the Stable Diffusion AI model includes only language and image decoder brain components. For instance, the Stable Diffusion model will pull NO WAR banner-bearers like this:

V1:

V2.1:

The shot shows text, although the model never learned to read or write. The model's string tokenizer automatically converts letters to lowercase before generating the image, so typing NO WAR banner or no war banner is the same.

I can also ask the model to draw a gorgeous woman:

V1:

V2.1:

The first image is gorgeous but physically incorrect. A second one is better, although it has an Uncanny valley feel. BTW, v2 has a lifehack to add a negative prompt and define what we don't want on the image. Readers might try adding horrible anatomy to the gorgeous woman request.

If we ask for a cartoon attractive woman, the results are nice, but accuracy doesn't matter:

V1:

V2.1:

Another example: I ordered a model to sketch a mouse, which looks beautiful but has too many legs, ears, and fingers:

V1:

V2.1: improved but not perfect.

V1 produces a fun cartoon flying mouse if I want something more abstract:

I tried multiple times with V2.1 but only received this:

The image is OK, but the first version is closer to the request.

Stable Diffusion struggles to draw letters, fingers, etc. However, abstract images yield interesting outcomes. A rural landscape with a modern metropolis in the background turned out well:

V1:

V2.1:

Generative models help make paintings too (at least, abstract ones). I searched Google Image Search for modern art painting to see works by real artists, and this was the first image:

“Modern art painting” © Google’s Image search result

I typed "abstract oil painting of people dancing" and got this:

V1:

V2.1:

It's a different style, but I don't think the AI-generated graphics are worse than the human-drawn ones.

The AI model cannot think like humans. It thinks nothing. A stable diffusion model is a billion-parameter matrix trained on millions of text-image pairs. I input "robot is creating a picture with a pen" to create an image for this post. Humans understand requests immediately. I tried Stable Diffusion multiple times and got this:

This great artwork has a pen, robot, and sketch, however it was not asked. Maybe it was because the tokenizer deleted is and a words from a statement, but I tried other requests such robot painting picture with pen without success. It's harder to prompt a model than a person.

I hope Stable Diffusion's general effects are evident. Despite its limitations, it can produce beautiful photographs in some settings. Readers who want to use Stable Diffusion results should be warned. Source code examination demonstrates that Stable Diffusion images feature a concealed watermark (text StableDiffusionV1 and SDV2) encoded using the invisible-watermark Python package. It's not a secret, because the official Stable Diffusion repository's test watermark.py file contains a decoding snippet. The put watermark line in the txt2img.py source code can be removed if desired. I didn't discover this watermark on photographs made by the online Hugging Face demo. Maybe I did something incorrectly (but maybe they are just not using the txt2img script on their backend at all).

Conclusion

The Stable Diffusion model was fascinating. As I mentioned before, trying something yourself is always better than taking someone else's word, so I encourage readers to do the same (including this article as well;).

Is Generative AI a game-changer? My humble experience tells me:

I think that place has a lot of potential. For designers and artists, generative AI can be a truly useful and innovative tool. Unfortunately, it can also pose a threat to some of them since if users can enter a text field to obtain a picture or a website logo in a matter of clicks, why would they pay more to a different party? Is it possible right now? unquestionably not yet. Images still have a very poor quality and are erroneous in minute details. And after viewing the image of the stunning woman above, models and fashion photographers may also unwind because it is highly unlikely that AI will replace them in the upcoming years.
Today, generative AI is still in its infancy. Even 768x768 images are considered to be of a high resolution when using neural networks, which are computationally highly expensive. There isn't an AI model that can generate high-resolution photographs natively without upscaling or other methods, at least not as of the time this article was written, but it will happen eventually.
It is still a challenge to accurately represent knowledge in neural networks (information like how many legs a cat has or the year Napoleon was born). Consequently, AI models struggle to create photorealistic photos, at least where little details are important (on the other side, when I searched Google for modern art paintings, the results are often even worse;).
When compared to the carefully chosen images from official web pages or YouTube reviews, the average output quality of a Stable Diffusion generation process is actually less attractive because to its high degree of randomness. When using the same technique on their own, consumers will theoretically only view those images as 1% of the results.

Anyway, it's exciting to witness this area's advancement, especially because the project is open source. Google's Imagen and DALL-E 2 can also produce remarkable findings. It will be interesting to see how they progress.

More on Technology

Nikhil Vemu

3 years ago

7 Mac Tips You Never Knew You Needed

Unleash the power of the Option key ⌥

#1 Open a link in the Private tab first.

Previously, if I needed to open a Safari link in a private window, I would:

copied the URL with the right click command,
choose File > New Private Window to open a private window, and
clicked return after pasting the URL.

I've found a more straightforward way.

Right-clicking a link shows this, right?

This, and all the images below are by the author

Hold option (⌥) for:

‘Open Link in New Private Window’ in Mac Safari

Click Open Link in New Private Window while holding.

Finished!

#2. Instead of searching for specific characters, try this

You may use unicode for business or school. Most people Google them when they need them.

That is lengthy!

You can type some special characters just by pressing ⌥ and a key.

For instance

• ⌥+2 -> ™ (Trademark)
• ⌥+0 -> ° (Degree)
• ⌥+G -> © (Copyright)
• ⌥+= -> ≠ (Not equal to)
• ⌥+< -> ≤ (Less than or equal to)
• ⌥+> -> ≥ (Greater then or equal to)
• ⌥+/ -> ÷ (Different symbol for division)

#3 Activate Do Not Disturb silently.

Do Not Disturb when sharing my screen is awkward for me (because people may think Im trying to hide some secret notifications).

Here's another method.

Hold ⌥ and click on Time (at the extreme right on the menu-bar).

Now, DND is activated (secretly!). To turn it off, do it again.

Note: This works only for DND focus.

#4. Resize a window starting from its center

Although this is rarely useful, it is still a hidden trick.

When you resize a window, the opposite edge or corner is used as the pivot, right?

However, if you want to resize it with its center as the pivot, hold while doing so.

#5. Yes, Cut-Paste is available on Macs as well (though it is slightly different).

I call it copy-move rather than cut-paste. This is how it works.

Carry it out.

Choose a file (by clicking on it), then copy it (⌘+C).

Go to a new location on your Mac. Do you use ⌘+V to paste it? However, to move it, press ⌘+⌥+V.

This removes the file from its original location and copies it here. And it works exactly like cut-and-paste on Windows.

#6. Instantly expand all folders

Set your Mac's folders to List view.

Assume you have one folder with multiple subfolders, each of which contains multiple files. And you wanted to look at every single file that was over there.

How would you do?

You're used to clicking the ⌄ glyph near the folder and each subfolder to expand them all, right? Instead, hold down ⌥ while clicking ⌄ on the parent folder.

This is what happens next.

Everything expands.

View/Copy a file's path as an added bonus

If you want to see the path of a file in Finder, select it and hold ⌥, and you'll see it at the bottom for a moment.

To copy its path, right-click on the folder and hold down ⌥ to see this

Click on Copy <"folder name"> as Pathname to do it.

#7 "Save As"

I was irritated by the lack of "Save As" in Pages when I first got a Mac (after 15 years of being a Windows guy).

It was necessary for me to save the file as a new file, in a different location, with a different name, or both.

Unfortunately, I couldn't do it on a Mac.

However, I recently discovered that it appears when you hold ⌥ when in the File menu.

Yay!

Sukhad Anand

3 years ago

How Do Discord's Trillions Of Messages Get Indexed?

They depend heavily on open source..

Discord users send billions of messages daily. Users wish to search these messages. How do we index these to search by message keywords?

Let’s find out.

Discord utilizes Elasticsearch. Elasticsearch is a free, open search engine for textual, numerical, geographical, structured, and unstructured data. Apache Lucene powers Elasticsearch.
How does elastic search store data? It stores it as numerous key-value pairs in JSON documents.
How does elastic search index? Elastic search's index is inverted. An inverted index lists every unique word in every page and where it appears.

4. Elasticsearch indexes documents and generates an inverted index to make data searchable in near real-time. The index API adds or updates JSON documents in a given index.

Let's examine how discord uses Elastic Search. Elasticsearch prefers bulk indexing. Discord couldn't index real-time messages. You can't search posted messages. You want outdated messages.

6. Let's check what bulk indexing requires.
1. A temporary queue for incoming communications.
2. Indexer workers that index messages into elastic search.

Discord's queue is Celery. The queue is open-source. Elastic search won't run on a single server. It's clustered. Where should a message go? Where?

8. A shard allocator decides where to put the message. Nevertheless. Shattered? A shard combines elastic search and index on. So, these two form a shard which is used as a unit by discord. The elastic search itself has some shards. But this is different, so don’t get confused.

Now, the final part is service discovery — to discover the elastic search clusters and the hosts within that cluster. This, they do with the help of etcd another open source tool.

A great thing to notice here is that discord relies heavily on open source systems and their base implementations which is very different from a lot of other products.

James Brockbank

3 years ago

Canonical URLs for Beginners

Canonicalization and canonical URLs are essential for SEO, and improper implementation can negatively impact your site's performance.

Canonical tags were introduced in 2009 to help webmasters with duplicate or similar content on multiple URLs.

To use canonical tags properly, you must understand their purpose, operation, and implementation.

Canonical URLs and Tags

Canonical tags tell search engines that a certain URL is a page's master copy. They specify a page's canonical URL. Webmasters can avoid duplicate content by linking to the "canonical" or "preferred" version of a page.

How are canonical tags and URLs different? Can these be specified differently?

Canonical URLs

What's a canonical link? Canonical link is the'master' URL for duplicate pages.

In Google's own words:

A canonical URL is the page Google thinks is most representative of duplicate pages on your site.
— Google Search Console Help

You can indicate your preferred canonical URL. For various reasons, Google may choose a different page than you.

When set correctly, the canonical URL is usually your specified URL.

Canonical URLs determine which page will be shown in search results (unless a duplicate is explicitly better for a user, like a mobile version).

Canonical URLs can be on different domains.

Other ways to specify canonical URLs

Canonical tags are the most common way to specify a canonical URL.

You can also set canonicals by:

Setting the HTTP header rel=canonical.
All pages listed in a sitemap are suggested as canonicals, but Google decides which pages are duplicates.
Redirects 301.

Google recommends these methods, but they aren't all appropriate for every situation, as we'll see below. Each has its own recommended uses.

Setting canonical URLs isn't required; if you don't, Google will use other signals to determine the best page version.

To control how your site appears in search engines and to avoid duplicate content issues, you should use canonicalization effectively.

Why Duplicate Content Exists

Before we discuss why you should use canonical URLs and how to specify them in popular CMSs, we must first explain why duplicate content exists. Nobody intentionally duplicates website content.

Content management systems create multiple URLs when you launch a page, have indexable versions of your site, or use dynamic URLs.

Assume the following URLs display the same content to a user:

A search engine sees eight duplicate pages, not one.

URLs #1 and #2: the CMS saves product URLs with and without the category name.
#3, #4, and #5 result from the site being accessible via HTTP, HTTPS, www, and non-www.
#6 is a subdomain mobile-friendly URL.
URL #7 lacks URL #2's trailing slash.
URL #8 uses a capital "A" instead of a lowercase one.

Duplicate content may also exist in URLs like:

https://www.website.com
https://www.website.com/index.php

Duplicate content is easy to create.

Canonical URLs help search engines identify different page variations as a single URL on many sites.

SEO Canonical URLs

Canonical URLs help you manage duplicate content that could affect site performance.

Canonical URLs are a technical SEO focus area for many reasons.

Specify URL for search results

When you set a canonical URL, you tell Google which page version to display.

Which would you click?

https://www.domain.com/page-1/

https://www.domain.com/index.php?id=2

First, probably.

Canonicals tell search engines which URL to rank.

Consolidate link signals on similar pages

When you have duplicate or nearly identical pages on your site, the URLs may get external links.

Canonical URLs consolidate multiple pages' link signals into a single URL.

This helps your site rank because signals from multiple URLs are consolidated into one.

Syndication management

Content is often syndicated to reach new audiences.

Canonical URLs consolidate ranking signals to prevent duplicate pages from ranking and ensure the original content ranks.

Avoid Googlebot duplicate page crawling

Canonical URLs ensure that Googlebot crawls your new pages rather than duplicated versions of the same one across mobile and desktop versions, for example.

Crawl budgets aren't an issue for most sites unless they have 100,000+ pages.

How to Correctly Implement the rel=canonical Tag

Using the header tag rel="canonical" is the most common way to specify canonical URLs.

Adding tags and HTML code may seem daunting if you're not a developer, but most CMS platforms allow canonicals out-of-the-box.

These URLs each have one product.

How to Correctly Implement a rel="canonical" HTTP Header

A rel="canonical" HTTP header can replace canonical tags.

This is how to implement a canonical URL for PDFs or non-HTML documents.

You can specify a canonical URL in your site's.htaccess file using the code below.

<Files "file-to-canonicalize.pdf"> Header add Link "< http://www.website.com/canonical-page/>; rel=\"canonical\"" </Files>

301 redirects for canonical URLs

Google says 301 redirects can specify canonical URLs.

Only the canonical URL will exist if you use 301 redirects. This will redirect duplicates.

This is the best way to fix duplicate content across:

HTTPS and HTTP
Non-WWW and WWW
Trailing-Slash and Non-Trailing Slash URLs

On a single page, you should use canonical tags unless you can confidently delete and redirect the page.

Sitemaps' canonical URLs

Google assumes sitemap URLs are canonical, so don't include non-canonical URLs.

This does not guarantee canonical URLs, but is a best practice for sitemaps.

Best-practice Canonical Tag

Once you understand a few simple best practices for canonical tags, spotting and cleaning up duplicate content becomes much easier.

Always include:

One canonical URL per page

If you specify multiple canonical URLs per page, they will likely be ignored.

Correct Domain Protocol

If your site uses HTTPS, use this as the canonical URL. It's easy to reference the wrong protocol, so check for it to catch it early.

Trailing slash or non-trailing slash URLs

Be sure to include trailing slashes in your canonical URL if your site uses them.

Specify URLs other than WWW

Search engines see non-WWW and WWW URLs as duplicate pages, so use the correct one.

Absolute URLs

To ensure proper interpretation, canonical tags should use absolute URLs.

So use:

<link rel="canonical" href="https://www.website.com/page-a/" />

And not:

<link rel="canonical" href="/page-a/" />

If not canonicalizing, use self-referential canonical URLs.

When a page isn't canonicalizing to another URL, use self-referencing canonical URLs.

Canonical tags refer to themselves here.

Common Canonical Tags Mistakes

Here are some common canonical tag mistakes.

301 Canonicalization

Set the canonical URL as the redirect target, not a redirected URL.

Incorrect Domain Canonicalization

If your site uses HTTPS, don't set canonical URLs to HTTP.

Irrelevant Canonicalization

Canonicalize URLs to duplicate or near-identical content only.

SEOs sometimes try to pass link signals via canonical tags from unrelated content to increase rank. This isn't how canonicalization should be used and should be avoided.

Multiple Canonical URLs

Only use one canonical tag or URL per page; otherwise, they may all be ignored.

When overriding defaults in some CMSs, you may accidentally include two canonical tags in your page's <head>.

Pagination vs. Canonicalization

Incorrect pagination can cause duplicate content. Canonicalizing URLs to the first page isn't always the best solution.

Canonicalize to a 'view all' page.

How to Audit Canonical Tags (and Fix Issues)

Audit your site's canonical tags to find canonicalization issues.

SEMrush Site Audit can help. You'll find canonical tag checks in your website's site audit report.

Let's examine these issues and their solutions.

No Canonical Tag on AMP

Site Audit will flag AMP pages without canonical tags.

Canonicalization between AMP and non-AMP pages is important.

Add a rel="canonical" tag to each AMP page's head>.

No HTTPS redirect or canonical from HTTP homepage

Duplicate content issues will be flagged in the Site Audit if your site is accessible via HTTPS and HTTP.

You can fix this by 301 redirecting or adding a canonical tag to HTTP pages that references HTTPS.

Broken canonical links

Broken canonical links won't be considered canonical URLs.

This error could mean your canonical links point to non-existent pages, complicating crawling and indexing.

Update broken canonical links to the correct URLs.

Multiple canonical URLs

This error occurs when a page has multiple canonical URLs.

Remove duplicate tags and leave one.

Canonicalization is a key SEO concept, and using it incorrectly can hurt your site's performance.

Once you understand how it works, what it does, and how to find and fix issues, you can use it effectively to remove duplicate content from your site.

Canonicalization SEO Myths

Wayne Duggan

3 years ago

What An Inverted Yield Curve Means For Investors

The yield spread between 10-year and 2-year US Treasury bonds has fallen below 0.2 percent, its lowest level since March 2020. A flattening or negative yield curve can be a bad sign for the economy.

What Is An Inverted Yield Curve?

In the yield curve, bonds of equal credit quality but different maturities are plotted. The most commonly used yield curve for US investors is a plot of 2-year and 10-year Treasury yields, which have yet to invert.

A typical yield curve has higher interest rates for future maturities. In a flat yield curve, short-term and long-term yields are similar. Inverted yield curves occur when short-term yields exceed long-term yields. Inversions of yield curves have historically occurred during recessions.

Inverted yield curves have preceded each of the past eight US recessions. The good news is they're far leading indicators, meaning a recession is likely not imminent.

Every US recession since 1955 has occurred between six and 24 months after an inversion of the two-year and 10-year Treasury yield curves, according to the San Francisco Fed. So, six months before COVID-19, the yield curve inverted in August 2019.

Looking Ahead

The spread between two-year and 10-year Treasury yields was 0.18 percent on Tuesday, the smallest since before the last US recession. If the graph above continues, a two-year/10-year yield curve inversion could occur within the next few months.

According to Bank of America analyst Stephen Suttmeier, the S&P 500 typically peaks six to seven months after the 2s-10s yield curve inverts, and the US economy enters recession six to seven months later.

Investors appear unconcerned about the flattening yield curve. This is in contrast to the iShares 20+ Year Treasury Bond ETF TLT +2.19% which was down 1% on Tuesday.

Inversion of the yield curve and rising interest rates have historically harmed stocks. Recessions in the US have historically coincided with or followed the end of a Federal Reserve rate hike cycle, not the start.

Bradley Vangelder

3 years ago

How we started and then quickly sold our startup

From a simple landing where we tested our MVP to a platform that distributes 20,000 codes per month, we learned a lot.

Starting point

Kwotet was my first startup. Everyone might post book quotes online.

I wanted a change.

Kwotet lacked attention, thus I felt stuck. After experiencing the trials of starting Kwotet, I thought of developing a waitlist service, but I required a strong co-founder.

I knew Dries from school, but we weren't close. He was an entrepreneurial programmer who worked a lot outside school. I needed this.

We brainstormed throughout school hours. We developed features to put us first. We worked until 3 am to launch this product.

Putting in the hours is KEY when building a startup

The instant that we lost our spark

In Belgium, college seniors do their internship in their last semester.

As we both made the decision to pick a quite challenging company, little time was left for Lancero.

Eventually, we lost interest. We lost the spark…

The only logical choice was to find someone with the same spark we started with to acquire Lancero.

And we did @ MicroAcquire.

Sell before your product dies. Make sure to profit from all the gains.

What did we do following the sale?

Not far from selling Lancero I lost my dad. I was about to start a new company. It was focused on positivity. I got none left at the time.

We still didn’t let go of the dream of becoming full-time entrepreneurs. As Dries launched the amazing company Plunk, and I’m still in the discovering stages of my next journey!

Dream!

You’re an entrepreneur if:

You're imaginative.
You enjoy disassembling and reassembling things.
You're adept at making new friends.
YOU HAVE DREAMS.

You don’t need to believe me if I tell you “everything is possible”… I wouldn't believe it myself if anyone told me this 2 years ago.

Until I started doing, living my dreams.

Vitalik

3 years ago

An approximate introduction to how zk-SNARKs are possible (part 2)

If tasked with the problem of coming up with a zk-SNARK protocol, many people would make their way to this point and then get stuck and give up. How can a verifier possibly check every single piece of the computation, without looking at each piece of the computation individually? But it turns out that there is a clever solution.

Polynomials

Polynomials are a special class of algebraic expressions of the form:

$x+5$
$x^4$
$x^3+3x^2+3x+1$
$628x^{271}+318x^{270}+530x^{269}+…+69x+381$

i.e. they are a sum of any (finite!) number of terms of the form $cx^k$

There are many things that are fascinating about polynomials. But here we are going to zoom in on a particular one: polynomials are a single mathematical object that can contain an unbounded amount of information (think of them as a list of integers and this is obvious). The fourth example above contained 816 digits of tau, and one can easily imagine a polynomial that contains far more.

Furthermore, a single equation between polynomials can represent an unbounded number of equations between numbers. For example, consider the equation $A(x)+ B(x) = C(x)$ . If this equation is true, then it's also true that:

$A(0)+B(0)=C(0)$
$A(1)+B(1)=C(1)$
$A(2)+B(2)=C(2)$
$A(3)+B(3)=C(3)$

And so on for every possible coordinate. You can even construct polynomials to deliberately represent sets of numbers so you can check many equations all at once. For example, suppose that you wanted to check:

$12+1=13$
$10+8=18$
$15+8=23$
$15+13=28$

You can use a procedure called Lagrange interpolation to construct polynomials $A(x)$ that give $(12,10,15,15)$ as outputs at some specific set of coordinates (eg. $(0,1,2,3)$ ), $B(x)$ the outputs $(1,8,8,13)$ on thos same coordinates, and so forth. In fact, here are the polynomials:

$A(x)=-2x^3+\frac{19}{2}x^2-\frac{19}{2}x+12$
$B(x)=2x^3-\frac{19}{2}x^2+\frac{29}{2}x+1$
$C(x)=5x+13$

Checking the equation $A(x)+B(x)=C(x)$ with these polynomials checks all four above equations at the same time.

Comparing a polynomial to itself

You can even check relationships between a large number of adjacent evaluations of the same polynomial using a simple polynomial equation. This is slightly more advanced. Suppose that you want to check that, for a given polynomial $F, F(x+2)=F(x)+F(x+1)$ with the integer range ${0,1\dots89}$ (so if you also check $F(0)=F(1)=1$ , then $F(100)$ would be the 100th Fibonacci number)

As polynomials, $F(x+2)-F(x+1)-F(x)$ would not be exactly zero, as it could give arbitrary answers outside the range $x={0,1\dots98}$ . But we can do something clever. In general, there is a rule that if a polynomial $P$ is zero across some set $S=\{x_1,x_2…x_n\}$ then it can be expressed as $P(x)=Z(x)*H(x)$ , where $Z(x)=(x-x_1)*(x-x_2)*\dots*(x-x_n)$ and $H(x)$ is also a polynomial. In other words, any polynomial that equals zero across some set is a (polynomial) multiple of the simplest (lowest-degree) polynomial that equals zero across that same set.

Why is this the case? It is a nice corollary of polynomial long division: the factor theorem. We know that, when dividing $P(x)$ by $Z(x)$ , we will get a quotient $Q(x)$ and a remainder $R(x)$ is strictly less than that of $Z(x)$ . Since we know that $P$ is zero on all of $S$ , it means that $R$ has to be zero on all of $S$ as well. So we can simply compute $R(x)$ via polynomial interpolation, since it's a polynomial of degree at most $n-1$ and we know $n$ values (the zeros at $S$ ). Interpolating a polynomial with all zeroes gives the zero polynomial, thus $R(x)=0$ and $H(x)=Q(x)$ .

Going back to our example, if we have a polynomial $F$ that encodes Fibonacci numbers (so $F(x+2)=F(x)+F(x+1)$ across $x=\{0,1…98\}$ ), then I can convince you that $F$ actually satisfies this condition by proving that the polynomial $P(x)=F(x+2)-F(x+1)-F(x)$ is zero over that range, by giving you the quotient:
$H(x)=\frac{F(x+2)-F(x+1)-F(x)}{Z(x)}$
Where $Z(x) = (x-0)*(x-1)*\dots*(x-98)$ .
You can calculate $Z(x)$ yourself (ideally you would have it precomputed), check the equation, and if the check passes then $F(x)$ satisfies the condition!

Now, step back and notice what we did here. We converted a 100-step-long computation into a single equation with polynomials. Of course, proving the N'th Fibonacci number is not an especially useful task, especially since Fibonacci numbers have a closed form. But you can use exactly the same basic technique, just with some extra polynomials and some more complicated equations, to encode arbitrary computations with an arbitrarily large number of steps.

see part 3

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